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A117936 Triangle, rows = inverse binomial transforms of A073133 columns. 4
1, 1, 1, 2, 3, 2, 3, 9, 12, 6, 5, 24, 56, 60, 24, 8, 62, 228, 414, 360, 120, 13, 156, 864, 2400, 3480, 2520, 720, 21, 387, 3132, 12606, 27360, 32640, 20160, 5040, 34, 951, 11034, 62220, 190704, 335160, 337680, 181440, 40320, 55, 2323, 38136, 294588, 1229760, 2997120, 4394880, 3820320, 1814400, 362880 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.

Note that binomial transforms are defined from offset 1 here. - R. J. Mathar, Aug 16 2019

LINKS

Table of n, a(n) for n=1..55.

FORMULA

Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.

EXAMPLE

First few columns of A073133 are: (1, 1, 1,...); (1, 2, 3,...); (2, 5, 10, 17,...); (3, 12, 33, 72,...). As sequences, these are f(x), Fibonacci polynomials: (1); (x); (x^2 + 1); (x^3 + 2x); (x^4 + 3x^2 + 1); (x^5 + 4x^3 + 3x);...For example, f(x), x = 1,2,3...using (x^4 + 3x^2 + 1) generates Column 5 of A073133: (5, 29, 109, 305...).

Inverse binomial transforms of the foregoing columns generates the triangle rows:

1;

1, 1;

2, 3, 2;

3, 9, 12, 6;

5, 24, 56, 60, 24;

8, 62, 228, 414, 360, 120;

...

MAPLE

A117936 := proc(n, k)

    add( A073133(i+1, n)*binomial(k-1, i)*(-1)^(i-k-1), i=0..k-1) ;

end proc:

seq(seq(A117936(n, k), k=1..n), n=1..13) ; # R. J. Mathar, Aug 16 2019

MATHEMATICA

(* A = A073133 *) A[_, 1] = 1; A[n_, k_] := A[n, k] = If[k < 0, 0, n A[n, k - 1] + A[n, k - 2]];

T[n_, k_] := Sum[A[i+1, n] Binomial[k-1, i] (-1)^(i - k - 1), {i, 0, k-1}];

Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 01 2020, from Maple *)

CROSSREFS

Cf. A073133, A117937, A117938, A006684 (column 2), A309717 (column 3 halved).

Sequence in context: A300663 A102310 A151546 * A264766 A251090 A078331

Adjacent sequences:  A117933 A117934 A117935 * A117937 A117938 A117939

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Apr 03 2006

STATUS

approved

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Last modified June 1 19:32 EDT 2020. Contains 334762 sequences. (Running on oeis4.)